%   ORIGINAL: h4/rich__list/EVERY__FOLDL__MAP
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/rich__list/FOLDL__MAP: !l g f e. h4/list/FOLDL f e (h4/list/MAP g l) = h4/list/FOLDL (\x y. f x (g y)) e l
% Assm: h4/rich__list/EVERY__FOLDL: !l P. h4/list/EVERY P l <=> h4/list/FOLDL (\l_27 x. l_27 /\ P x) T l
% Goal: !l P. h4/list/EVERY P l <=> h4/list/FOLDL $and T (h4/list/MAP P l)
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_richu_u_lists_FOLDLu_u_MAP]: !_1. (!f x g y. happ (happ (happ (happ _1 f) x) g) y = happ (happ f x) (happ g y)) ==> (!_0. (!f g x. happ (happ (happ _0 f) g) x = happ (happ (happ _1 f) x) g) ==> (!l g f e. h4/list/FOLDL f e (h4/list/MAP g l) = h4/list/FOLDL (happ (happ _0 f) g) e l))
% Assm [h4s_richu_u_lists_EVERYu_u_FOLDL]: !_1. (!l_27 P x. happ (happ (happ _1 l_27) P) x <=> l_27 /\ happ P x) ==> (!_0. (!P l_27. happ (happ _0 P) l_27 = happ (happ _1 l_27) P) ==> (!l P. h4/list/EVERY P l <=> h4/list/FOLDL (happ _0 P) T l))
% Goal: !l P. h4/list/EVERY P l <=> h4/list/FOLDL $and T (h4/list/MAP P l)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q159858,TV_Q159854]: ![V_f, V_g]: (![V_x]: s(TV_Q159854,happ(s(t_fun(TV_Q159858,TV_Q159854),V_f),s(TV_Q159858,V_x))) = s(TV_Q159854,happ(s(t_fun(TV_Q159858,TV_Q159854),V_g),s(TV_Q159858,V_x))) => s(t_fun(TV_Q159858,TV_Q159854),V_f) = s(t_fun(TV_Q159858,TV_Q159854),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_richu_u_lists_FOLDLu_u_MAP, axiom, ![TV_u_27b,TV_u_27a,TV_u_27c]: ![V_uu_1]: (![V_f, V_x, V_g, V_y]: s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,TV_u_27a)))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f))),s(TV_u_27a,V_x))),s(t_fun(TV_u_27c,TV_u_27b),V_g))),s(TV_u_27c,V_y))) = s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,happ(s(t_fun(TV_u_27c,TV_u_27b),V_g),s(TV_u_27c,V_y))))) => ![V_uu_0]: (![V_f, V_g, V_x]: s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27a)))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f))),s(t_fun(TV_u_27c,TV_u_27b),V_g))),s(TV_u_27a,V_x))) = s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),t_fun(TV_u_27a,t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27c,TV_u_27a)))),V_uu_1),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f))),s(TV_u_27a,V_x))),s(t_fun(TV_u_27c,TV_u_27b),V_g))) => ![V_l, V_g, V_f, V_e]: s(TV_u_27a,h4s_lists_foldl(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f),s(TV_u_27a,V_e),s(t_h4s_lists_list(TV_u_27b),h4s_lists_map(s(t_fun(TV_u_27c,TV_u_27b),V_g),s(t_h4s_lists_list(TV_u_27c),V_l))))) = s(TV_u_27a,h4s_lists_foldl(s(t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27a)),happ(s(t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27a))),happ(s(t_fun(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),t_fun(t_fun(TV_u_27c,TV_u_27b),t_fun(TV_u_27a,t_fun(TV_u_27c,TV_u_27a)))),V_uu_0),s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27a)),V_f))),s(t_fun(TV_u_27c,TV_u_27b),V_g))),s(TV_u_27a,V_e),s(t_h4s_lists_list(TV_u_27c),V_l)))))).
fof(ah4s_richu_u_lists_EVERYu_u_FOLDL, axiom, ![TV_u_27a]: ![V_uu_1]: (![V_lu_27, V_P, V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_bool,t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool))),V_uu_1),s(t_bool,V_lu_27))),s(t_fun(TV_u_27a,t_bool),V_P))),s(TV_u_27a,V_x)))) <=> (p(s(t_bool,V_lu_27)) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))))) => ![V_uu_0]: (![V_P, V_lu_27]: s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_bool,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_bool,t_fun(TV_u_27a,t_bool))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_bool,V_lu_27))) = s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_bool,t_fun(t_fun(TV_u_27a,t_bool),t_fun(TV_u_27a,t_bool))),V_uu_1),s(t_bool,V_lu_27))),s(t_fun(TV_u_27a,t_bool),V_P))) => ![V_l, V_P]: s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_bool,h4s_lists_foldl(s(t_fun(t_bool,t_fun(TV_u_27a,t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_bool,t_fun(TV_u_27a,t_bool))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_P))),s(t_bool,t),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
fof(ch4s_richu_u_lists_EVERYu_u_FOLDLu_u_MAP, conjecture, ![TV_u_27a]: ![V_l, V_P]: s(t_bool,h4s_lists_every(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_bool,h4s_lists_foldl(s(t_fun(t_bool,t_fun(t_bool,t_bool)),d_and),s(t_bool,t),s(t_h4s_lists_list(t_bool),h4s_lists_map(s(t_fun(TV_u_27a,t_bool),V_P),s(t_h4s_lists_list(TV_u_27a),V_l)))))).
